1. Field of the Invention
The following invention relates to sub-diffraction limit resolution in microscopy. The invention has particular utility in the use of microscopy in the testing of fracture toughness of thin ceramic substrates and will be described in connection with such utility, although other utilities such as measuring sub-micron size particles including biological particles.
2. Description of the Prior Art
Indentation techniques are well developed for hardness study. The American Society of Testing and Materials (ASTM) developed a standard test method for Vickers indentation hardness of advanced ceramics (ASTM C 1327-96a, 1996) incorporated herein by reference. Vickers indentation techniques have also been widely used for studying fracture toughness of brittle materials such as glass and ceramics since surface crack traces were first recognized as indicative of fracture toughness by Palmqvist in 1957. These crack traces are referred to as indention traces or Palmqvist cracks.
In general, the procedure of the Vickers indentation toughness test includes producing an indentation on a plane surface of the material under investigation by a standard hardness tester and subsequently studying the induced cracks by a microscope. It is important to note that indentation is considered micro when the applied indenter load is less than 5N, otherwise, indentation is called macro indentation.
With the measured data of the indenter, load, and the dimensions of the induced cracks, it is possible to evaluate the toughness of the material. For example, a Vickers hardness tester usually makes a diamond indentation with cracks emanating from the diamond corners as shown in FIG. 1. For most mathematical models based on the Vickers hardness tester and published in the literature, the c/a or l/a ratio depicted in FIG. 1 was limited to a certain range. For example, Niihara et al (1982) proposed an equation that requires the 1/a ratio to be between 0.25 and 2.5.
The advantages of the Vickers indentation toughness technique are the simplicity and cost effectiveness of the measurement procedure. The specimen preparation is also relatively simple, requiring only a flat surface. And, at least 10 tests can be performed on a surface of only 100 mm2. The disadvantage of this technique is that an accurate measurement of the crack length c or l, usually measured under an optical microscope, is difficult. The indentation induced cracks are often hard, if not impossible, to observe because the width of indentation-induced cracks is very narrow, especially near the crack tips that the indention-induced cracks are beyond the resolution of common optical microscopes. Although measurements of the indention induced cracks can be conducted under a scanning electronic microscope (SEM), the usage of a SEM will significantly slow down the experimental procedure and greatly increase experiment costs.
Also, ordinary optical microscopes are limited in resolving power, and therefore cannot observe smaller indention cracks using light diffraction. Even assuming an optical system is perfect, because of the wave property of the light, the smallest spot resolvable by an optical microscope is ultimately defined by the diffraction of the illuminating light. At a small enough scale, physical optics principles take effect, i.e., the wave-like motion of light will deflect around comers of an object under observation to a tiny but finite degree. This phenomenon is known as the “diffraction limit” of an optical microscope. For example, suppose two point sources of light are to be imaged by a microscope. Because of light diffraction the two point sources of light will be imaged by a microscope as two discs of light distribution. These discs are each referred to as an Airy Disc, i.e., a high irradiance circular spot. FIG. 2 shows graphically a light distribution pattern of an Airy Disc of a point source due to light diffracting from an object under observation.
As shown in FIG. 2, the Airy Disc consists of a central bright peak surrounded by a set of concentric dark and light rings. The resolution limit of a microscope is defined as the distance of the two point sources at which their images has a separation so that the peak of one Airy Disc coincides with the first dark ring of the other. This is referred to as the Rayleigh's Criterion for resolution. The numerical expression of Rayleigh's Criterion is as follows:
                    d        =                              1.22            ⁢                                          λ                ⁢                                                                  ⁢                f                            D                                =                      0.61            ⁢                          λ                              N                .                A                .                                                                        (        1        )            where d is the smallest distance between two objects resolvable by a microscope, λ is the wavelength of light, f is the focal length of the microscope's objective lens, D is the diameter of the aperture of the microscope, and N.A. is the numerical aperture of the microscope (Smith, 1966).
Using Eq. (1), a numerical value of the resolution imposed by the diffraction limit can be calculated. For example, for a modem microscope objective lens having a N.A. of 1.3, assuming that the illumination light has a wavelength of 400 nm, the smallest object the microscope can resolve is 200 nm. However, it is desirable to be able to optically observe objects smaller than that scale.
Several designs have been invented to overcome the aforementioned problem with microscopes available in the art. Among them are confocal microscopes with a spatial resolution of 200 nm (Pawley, 1995), and near-field scanning microscopes with a spatial resolution of 60 nm (Dunn, R. C., 1999). There is also an older technique in optical microscopy called dark-field microscope, which is capable of observing particles of the size as small as 5 nm (Monk, 1963).
Outside the field of microscopy, there also exist several ways to observe structures with dimensions smaller than the diffraction limited scale. In optical testing, a Foucault knife-edge method is commonly used to find defects as small as one tenth of the wavelength λ/10 (e.g. 40 nm, using blue light illumination at 400 nm) in an optical component, such as a mirror surface. In this technique, an illuminated pinhole and a sharp knife-edge are located in the same plane away from the mirror (e.g. a spherical concave mirror) being tested. If the mirror surface is perfectly spherical and free of any defect, then an image of the pinhole will be formed with a uniform light distribution. When the knife-edge is moved across the line of light at the image point, a uniform shadow can be observed to cross the surface of the mirror. However, if very small surface defects exist on the mirror, these defects will cause the light impinged upon them to diffract and subsequently deform the spherical wave of the incident light. Now an observer behind the knife-edge will see light spots (diffraction patterns from the defects) on the dark shadow when the knife-edge is moved across the field (Longhurst, 1973). This technique resembles the method used in dark-field microscope, in which the direct illuminating light beam is obstructed and only half of the diffraction orders from the small particles are observed. Furthermore, an extension of the Foucault knife-edge, or the Schlieren method, is used to detect small variation of refractive index in a medium. The Schlieren method has been applied to fluid dynamics to study the behavior of a moving fluid (Longhurst. 1973).
In addition, to solve some of the above problems with microscopy, some researchers focused on the observability of indention cracks. Ponton and Rawlings (Ponton and Rawlings, 1989b) proposed a method where a minimum indenter load of about 50 N produces visible cracks so that accurate measurement of the indention cracks under common optical microscope. These macro-hardness testers have dominated the art because they ensure cracks produced by the Vickers hardness tester could be measured, and micro indentation was believed to produce no indentation cracking (Anton and Subhash, 2000). Other researchers have focused on improving the observability of indentation cracks produced using Vickers hardness testers by polishing the surface of the test specimens. The specimen surfaces were usually polished to at least 1 μm diamond finish (Ponton and Rawlings, 1989b). Although Ponton and Rawlings pointed out that processes such as polishing, could produce residual stresses on the surface to prevent correct test results (Ponton and Rawlings, 1989b), polishing seemed to be a necessary process for specimen preparation reported in the literature.
However, most of the prior art mathematical models are based on the assumption that there are no pre-existing surface stresses on test specimens. Although proper heat treatment could remove the stresses created by polishing; it may change the physical properties of the test specimen. Other prior art methods proposed to deal with the problems associated with these pre-existing stresses on specimens by highlighting the pre-existing surface cracks using a fluorescent dye penetrant (Ponton and Rawlings, 1989). However, these methods produce side effects, such as extra post-indentation slow crack growth in many ceramics, thereby preventing an accurate evaluation of the specimen's toughness.
There are other problems with the above mentioned methods of indention testing. Thin ceramic substrates are widely used as electrolytes in solid oxide electrolyzers, and are typically made by a tape-cast process. After sintering, the products are usually in the form of thin sheets with a typical thickness 0.5 mm or less in engineering applications. As a result, indenter load of 50 N tends to break the specimen substrates. In practice, the majority of the ceramic substrates with this thickness can only be indented by micro-indentation.
Other researchers in the art, Cook and Pharr (1990), found that a radial crack forms extremely early (possibly almost instantly) in the loading process (typically 0.8 N). Small cracks caused by such loads can not possibly be detected by the conventional optical methods described above. In addition, many thin ceramic substrates are used with an as-fired surface finish. Polishing of such surfaces would alter the actual fracture toughness of the substrates. However, leaving the surface of the substrate unpolished introduces even more difficulties in the observation and measurement of small cracks.
Thus, a better technique for measuring indentation cracks in thin substrates is needed.